Theorem flimclslem 24122

Description: Lemma for flimcls 24123. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
flimcls.2	⊢ 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))

Assertion

Ref	Expression
flimclslem	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))

Proof of Theorem flimclslem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		flimcls.2	. . 3 ⊢ 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
2		topontop 23051	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3	2	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
4		eqid 2763	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	neisspw 23245	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 ∪ 𝐽)
6	3, 5	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 ∪ 𝐽)
7		toponuni 23052	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
8	7	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 = ∪ 𝐽)
9	8	pweqd 4580	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
10	6, 9	sseqtrrd 3975	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝑋)
11		toponmax 23064	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
12		elpw2g 5305	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
13	11, 12	syl 18	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
14	13	biimpar 482	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
15	14	3adant3 1150	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
16	15	snssd 4753	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ⊆ 𝒫 𝑋)
17	10, 16	unssd 4146	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋)
18		ssun2 4133	. . . . . 6 ⊢ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})
19	11	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 ∈ 𝐽)
20		simp2 1155	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
21	19, 20	ssexd 5296	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
22	21	snn0d 4742	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ≠ ∅)
23		ssn0 4363	. . . . . 6 ⊢ (({𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∧ {𝑆} ≠ ∅) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
24	18, 22, 23	sylancr 598	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
25	20, 8	sseqtrd 3974	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
26		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ ((cls‘𝐽)‘𝑆))
27	4	neindisj 23255	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ (𝐴 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥 ∩ 𝑆) ≠ ∅)
28	27	expr 461	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥 ∩ 𝑆) ≠ ∅))
29	3, 25, 26, 28	syl21anc 850	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥 ∩ 𝑆) ≠ ∅))
30	29	imp 411	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑥 ∩ 𝑆) ≠ ∅)
31		elsni 4607	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑆} → 𝑦 = 𝑆)
32	31	ineq2d 4174	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑆} → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝑆))
33	32	neeq1d 3017	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑆} → ((𝑥 ∩ 𝑦) ≠ ∅ ↔ (𝑥 ∩ 𝑆) ≠ ∅))
34	30, 33	syl5ibrcom 250	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑦 ∈ {𝑆} → (𝑥 ∩ 𝑦) ≠ ∅))
35	34	ralrimiv 3156	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → ∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅)
36	35	ralrimiva 3157	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅)
37		simp1 1154	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ (TopOn‘𝑋))
38	4	clsss3 23197	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
39	3, 25, 38	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
40	39, 26	sseldd 3939	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ ∪ 𝐽)
41	40, 8	eleqtrrd 2866	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ 𝑋)
42	41	snssd 4753	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ⊆ 𝑋)
43		snnzg 4741	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑆) → {𝐴} ≠ ∅)
44	43	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ≠ ∅)
45		neifil 24018	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
46	37, 42, 44, 45	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
47		filfbas 23986	. . . . . . . 8 ⊢ (((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
48	46, 47	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
49		ne0i 4295	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑆) → ((cls‘𝐽)‘𝑆) ≠ ∅)
50	49	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
51		cls0 23218	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
52	3, 51	syl 18	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘∅) = ∅)
53	50, 52	neeqtrrd 3032	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅))
54		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑆 = ∅ → ((cls‘𝐽)‘𝑆) = ((cls‘𝐽)‘∅))
55	54	necon3i 2990	. . . . . . . . 9 ⊢ (((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅) → 𝑆 ≠ ∅)
56	53, 55	syl 18	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ≠ ∅)
57		snfbas 24004	. . . . . . . 8 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅ ∧ 𝑋 ∈ 𝐽) → {𝑆} ∈ (fBas‘𝑋))
58	20, 56, 19, 57	syl3anc 1398	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ∈ (fBas‘𝑋))
59		fbunfip 24007	. . . . . . 7 ⊢ ((((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋) ∧ {𝑆} ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅))
60	48, 58, 59	syl2anc 595	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅))
61	36, 60	mpbird 260	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
62		fsubbas 24005	. . . . . 6 ⊢ (𝑋 ∈ 𝐽 → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
63	19, 62	syl 18	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
64	17, 24, 61, 63	mpbir3and 1361	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋))
65		fgcl 24016	. . . 4 ⊢ ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
66	64, 65	syl 18	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
67	1, 66	eqeltrid 2867	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐹 ∈ (Fil‘𝑋))
68		fvex 6896	. . . . . 6 ⊢ ((nei‘𝐽)‘{𝐴}) ∈ V
69		snex 5412	. . . . . 6 ⊢ {𝑆} ∈ V
70	68, 69	unex 7744	. . . . 5 ⊢ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V
71		ssfii 9380	. . . . 5 ⊢ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
72	70, 71	ax-mp 5	. . . 4 ⊢ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
73		ssfg 24010	. . . . . 6 ⊢ ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
74	64, 73	syl 18	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
75	74, 1	sseqtrrdi 3979	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ 𝐹)
76	72, 75	sstrid 3949	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝐹)
77		snssg 4750	. . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
78	21, 77	syl 18	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
79	18, 78	mpbiri 261	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
80	76, 79	sseldd 3939	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ 𝐹)
81	76	unssad 4147	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
82		elflim 24109	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
83	37, 67, 82	syl2anc 595	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
84	41, 81, 83	mpbir2and 725	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ (𝐽 fLim 𝐹))
85	67, 80, 84	3jca 1146	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  {csn 4590  ∪ cuni 4873  ‘cfv 6538  (class class class)co 7412  ficfi 9371  fBascfbas 21491  filGencfg 21492  Topctop 23031  TopOnctopon 23048  clsccl 23156  neicnei 23235  Filcfil 23983   fLim cflim 24072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1o 8454  df-2o 8455  df-en 8945  df-fin 8948  df-fi 9372  df-fbas 21500  df-fg 21501  df-top 23032  df-topon 23049  df-cld 23157  df-ntr 23158  df-cls 23159  df-nei 23236  df-fil 23984  df-flim 24077

This theorem is referenced by:  flimcls  24123